Code-Carrier Divergence Monitoring for the GPS Local Area Augmentation System

Transcription

1 Coe-Carrier Divergence Monitoring for the GPS Local Area Augmentation System Dwaraanath V. Simili an Boris Pervan, Illinois Institute of Technology, Chicago, IL Abstract Coe-carrier smoothing is a commonly use metho in Differential GPS (DGPS) systems to mitigate the effects of receiver noise an multipath. The FAA s Local Area Augmentation System (LAAS) uses this technique to help provie the navigation performance neee for aircraft precision approach an laning. However, unless the reference an user smoothing filter implementations are carefully matche, ivergence between the coe an carrier ranging measurements will cause ifferential ranging errors. The FAA s LAAS Groun Facility (LGF) reference station will implement a prescribe first-orer Linear Time Invariant (LTI) filter. Yet flexibility must be provie to avionics manufacturers in their airborne filter implementations. While the LGF LTI filter is one possible means for airborne use, its relatively slow transient response (acceptable for a groun base receiver) is not ieal at the aircraft because of frequent filter resets following losses of low elevation satellite signals (cause by aircraft attitue motion). However, in the presence of a coe-carrier ivergence (CCD) anomaly at the GPS satellite, large ivergence rates are theoretically possible, an therefore protection must be provie by the LGF through irect monitoring for such events. In response, this paper aresses the impact of the CCD threat to LAAS ifferential ranging error an efines an LGF monitor to ensure navigation integrity. Differential ranging errors resulting from unmatche filter esigns an ifferent groun/air filter start times are analyze in etail, an the requirements for the LGF CCD monitor are erive. A CCD integrity monitor algorithm is then evelope to irectly estimate an etect anomalous ivergence rates. The monitor algorithm is implemente an successfully teste using archive fiel ata from the LAAS Test Prototype (LTP) at the William J. Hughes FAA Technical Center. Finally, the paper provies recommenations for initial monitor implementation an future wor. INTRODUCTION The Feeral Aviation Aministration (FAA) an the aviation inustry are currently eveloping the Local Area Augmentation System (LAAS). This is a ifferential GPS system that augments GPS navigation in two ways. First, the LAAS Groun Facility (LGF) provies ifferential corrections to the user (aircraft) that augment user accuracy. Secon, the LGF monitors the ranging sources to protect against faults that coul result in navigation errors, thereby ensuring the integrity of navigation for LAAS users. In orer to ensure system integrity, for each class of ranging source (i.e., satellite) fault there is a monitor at the LGF that computes a test metric esigne to inicate the presence of that particular type of fault. The purpose of these monitors is to limit the LAAS user s integrity ris ue to a fault occurring at the ranging source. LAAS ranging source faults are ivie into two categories []: an aitive bias to the ifferential correction error with no change in variance of the error, an an increase in variance of the ifferential correction error with no change in bias. Faults that result in the first in of error are Coe-Carrier Divergence (CCD), signal eformation, erroneous ephemeris, an excessive acceleration. A satellite low power fault results in an increase in the ranging error variance leaing to a fault of the secon type. In this paper we aress the coe-carrier ivergence fault. Divergence can be cause by ionospheric activity (nominal or storm) or a fault occurring at the ranging source (satellite). The latter is the primary CCD threat, as large ivergence rates are theoretically possible. This motivates the nee for CCD monitoring to be provie by the LGF. During ionospheric activity, the CCD monitor can also provie benefit by helping to etect moving storm fronts. However, the LGF has other monitors esignate to etect ionospheric anomalies. Lie other DGPS architectures, the LGF (reference station) an aircraft (user) employ coe-carrier smoothing to mitigate the effects of receiver noise an multipath. If the aircraft an groun implement ientical filter esigns an have the same start times, then they experience the same transient response to ivergence, an ifferential-ranging errors will not exist. However, if they have unmatche filter esigns or ifferent start times, then ivergence between coe an carrier ranging measurements will cause ifferential ranging errors. The LGF reference station will implement an FAA prescribe first-orer Linear Time Invariant (LTI) filter. One approach to mitigate the CCD threat is to have the groun an aircraft implement ientical smoothing filters. The cost, however, is a loss of esign flexibility for avionics manufactures, which is not esirable. Even though the LGF LTI filter is one possible means for airborne use, its relatively slow transient response (acceptable for a groun base receiver) is not ieal at the aircraft because the aircraft filter is liely to experience frequent filter resets following the loss of signals from low elevation satellites (cause by aircraft attitue motion). In this /06/$0.00/ 006 IEEE 483

2 paper we will consier various possible aircraft filter implementations an show that a first orer Linear Time Varying (LTV) filter is a goo choice for airborne implementation. The CCD monitor algorithm escribe in this paper has two functions: ivergence rate estimation an etection. Monitor threshols are establishe with the ai of LAAS Test Prototype (LTP) ata. For the integrity analysis we present a new irect approach for computing the overall probability of Loss Of Integrity P(LOI) an escribe a preliminary analysis for time to alert. The paper is ivie into five sections. Section I escribes the relevant requirements for CCD fault etection. Section II iscusses the ifferent types of possible smoothing filters that can be implemente at the aircraft. Section III efines the CCD monitor an provies experimental valiation using archive fiel ata from the LAAS Test Prototype (LTP) at the William J. Hughes FAA Technical Center an LGF test ata provie by Honeywell. Section IV presents the general integrity analysis for a space segment failure with application to CCD monitoring. Finally, Section V gives the conclusion. Ι. REQUIREMENTS This section briefly summarizes the relevant LGF specifications an requirements on ranging source integrity, groun monitor continuity, an the prescribe LGF smoothing filter algorithm. The relevant Minimal Operational Performance Stanars (MOPS) avionics requirements are also iscusse. Further etails on requirements can be foun in [4, 5]. A. LGF Category I Integrity Requirement: The probability that the LGF transmits Misleaing Information (MI) for 3 secons or longer ue to a Ranging Source (RS) failure shall not excee uring any 150 sec approach interval. MI is efine as broacast ata that results in the lateral or vertical position error exceeing protection levels for any user w/in 60 nmi of the LGF. The CCD failure rate is efine to be 10-4 /hr for satellite (SV) acquisition, an the prior probability of CCD failure after acquisition is given as /SV/approach. B. LGF Reference Receiver (RR) an Groun Monitor Continuity Requirement: It is require that the probability of any vali ranging source is mae unavailable ue to a false alarm shall not excee per 15 sec interval. C. Requirement Allocations: For the purposes of this wor, we assume that the relevant allocations (from items A an B above) for the CCD monitor are: The probability of MI given a CCD fault is 10-4 per 150 sec approach interval, an the probability of fault free alarm is 10-7 per 15 sec interval. D. LGF Smoothing Filter Algorithm: The smoothing filter is efine in the LGF specification as follows. In steay state, each pseuorange measurement from each RS shall be smoothe using the filter: 1 N 1 PRs = PRr () + [PRs( 1) + φ() + φ( 1)] N N N = S T where, PR r = raw pseuorange, PR s = the smoothe pseuorange, S= time filter constant, equal to 100 secons, T = filter sample interval, nominally equal to 0.5 secons, φ= accumulate phase measurement, = current measurement, an -1 = previous measurement. The LGF is also require to generate an broacast to LAAS users σ pr_gn, the stanar eviation of the error on the broacast smoothe pseuorange correction. The variable σ pr_gn can be expresse as the root sum square (RSS) of the stanar eviation of groun ranging error ue to all sources except filter transient to nominal ionospheric ivergence (σ pr_gn,nom ) an the stanar eviation that accounts for groun transient filter responses to nominal ionospheric ivergence (σ iv_gn ). The nominal ionospheric CCD rate is given in LGF Specification as normally istribute with zero mean an stanar eviation of m/s. E. MOPS Avionics Requirements: The airborne system is also require to o carrier smoothing of the pseuorange measurements, but a specific filter is not efine. (This iffers from the LGF Specification, which prescribes the groun system filter implementation). The MOPS provies significant flexibility to avionics manufacturers by specifying that the airborne filter nee only match the groun filter with the following performance requirement: In response to a coe-carrier ivergence rate of up to m/s, the smoothing filter output shall achieve an error less than 0.5 m within 00 sec after initialization relative to the steay-state response of [filters specifie in LGF Spec]. Lie the groun system, the aircraft also has a σ pr_air which can be expresse as the RSS of σ pr_air,nom (stanar eviation of air ranging error ue to all sources except filter transient to nominal ionospheric ivergence) an σ iv_air (stanar eviation of air ranging error ue to filter transient to nominal ionospheric ivergence). The MOPS states that the aircraft must account for its filter transient response to nominal ivergence (i.e., after filter startup or reset) by inflating the stanar eviation use for ranging measurements in the erivation of its protection levels. This effect is capture by σ iv_air. 484

3 The MOPS further requires that the steay state value of σ iv_air shall not excee 0.15 m, an it states that steay state operation is efine to be following 360 secons of continuous operation of the smoothing filter. The relevant MOPS requirements can be summarize in graphical form as shown in Figure 1. The aircraft is permitte to have transient response to ivergence of m/s anywhere in the unshae region. The response of the LGF filter (one acceptable choice at the aircraft) is shown in the Figure 1. ±0.5 m ±0.15 m 360 sec Figure 1. MOPS-Compliant Transient Response II. GENERAL AIRCRAFT FILTERS Form Figure 1, it is clear if the orer an time-invariance of the avionics filter are unconstraine, there exists flexibility in MOPS-compliant transient response to ivergence inputs. Unfortunately, any significant variation from the LGF filter response is problematic (even if permitte by the MOPS) because it will result in a ifferential ranging error. The aircraft must account for any such eviation in σ iv_air, which increases σ pr_air as it is efine earlier i.e. RSS (σ pr_air_nom, σ iv_air,) with σ pr_air_nom being the nominal value of σ pr_air prior to accounting for ionosphere ivergence effects at the aircraft. But increasing σ pr_air can reuce system availability. Thus, in orer to maximize system availability, it is reasonable to assume that the goal of the avionics manufacturer is to eep σ pr_air small at any given time. For ivergence inputs, this means that minimizing overshoot an steay-state error will lower σ pr_air. For nominal coe noise inputs, a time varying filter implementation (at filter start-up) can provie quicer noise response, which will lower σ pr_air_nom an therefore σ pr_air. So these types of filters nee to be consiere. In this section, we show practical first an secon orer avionics filter esigns that are potentially useful. Figures (a) through Figure () show the transient responses of 1 st orer LTI an LTV MOPS-compliant aircraft filter implementations. Figure (a) shows the sample response to a white coe noise input (σ = 0.5 m) for the LGF LTI filter along with the theoretical 1-sigma error envelope. Figure (b) shows the response of the same filter to a nominal ivergence input. The slow transient response of this filter to noise will affect the aircraft more seriously than the groun because of frequent filter resets at the aircraft. Using a time varying gain in the first orer filter at the aircraft can significantly spee up the transient response to noise. Figure (c) shows a sample noise response using a filter with time constant τ = t for t < 100 sec an τ = 100 sec for t 100 sec. The rapi convergence of the error is reaily apparent as compare to the LTI case. Furthermore, the filter is simple to implement as it eviates from the LGF LTI filter only in that the gains are changing in the first 100 sec. The response of the LTV filter to a nominal ivergence input of m/s, shown in Figure (), suggests that the MOPS ivergence response requirements are not met (ashe lower tolerable limit line). However, when the MOPS efinition of steay state (360 sec) is use to efine the lower transient response bounary (soli line), the time response becomes acceptable. From the point of view of LGF CCD monitor esign the LTV filter must therefore be treate as a realistic caniate for airborne implementation. The use of secon orer filters at the aircraft can also be MOPS compliant. Figure (e) an Figure (f) show the noise an ivergence responses of a n orer LTI filter, which was esigne to simultaneously maximize overshoot (subject to the MOPS ivergence response requirements in Figure 1) an yet prouce negligible steay state error relative to the LGF implementation. As with the first orer case, secon orer LTV realizations are also possible to spee up noise response. While there is no clear time response benefit in the use of n orer filters in the avionics relative to 1 st orer LTV filter, the noise output of the n orer filter oes have less high frequency content. This may be beneficial to a navigation avionics manufacturer who esires to prouce smoother position inputs to the autopilot. Although, n orer filters o not help in lowering σ pr_air, the potential benefit of lowering the high-frequency content of the output means that the potential for airborne implementation of such a filter cannot be ignore. The overshoot exhibite in Figure (f) is not a necessary feature of a secon orer implementation, but it oes efine the MOPS compliant upper limit on the transient response for such filters. The overshoot in this n orer response woul cause a temporary increase in σ pr_air, (relative to a 1 st orer implementation) which may be acceptable to an avionics manufacturer. In any case, however, it is reasonable to assume that the aircraft woul not intentionally implement a filter with a steay state error ifferent from the reference LGF filter, as it woul cause an unnecessary increase in σ pr_air for the entire satellite pass therefore cause a potential availability penalty. As iscusse earlier, the MOPS gives enough room to implement a secon orer LTI/LTV filter. However, for the 485

4 (a) (b) (c) () (e) (f) rest of the analysis in this paper we assume that the aircraft uses a first orer LTV filter for smoothing. The implication on results if a secon orer filter is use can be consiere for future wor. In the next section, we present the algorithm evelopment of the CCD monitor. ΙII. CCD Monitor Algorithm The LGF ivergence monitor consists of two components: a ivergence rate estimator an a etection test. The input to the ivergence rate estimator, z, is the raw coe minus carrier measurement. The ivergence rate estimator ifferentiates the input z an filters the result using two first orer LTI filters in series (secon orer filter with real poles) to reuce the coe noise contribution to the estimation error. The continuoustime realization of the estimator algorithm (Laplace Transform) is, ˆ s D( s) = Z( s) (1) ( τ s + 1)( τ s 1) 1 + The estimator output ( ˆ in time omain) is the filtere ivergence estimate. The iscrete time equations for the rate estimator are given below with () being the final rate estimate: τ 1 ( ) 1 T 1 = 1( 1) + [ z( ) z( 1)] τ 1 τ 1 () τ T T ( ) = ( 1) + 1( 1) τ τ The filter time constants τ 1 an τ are algorithm parameters, whose values are nominally set at τ 1 = τ = 30 sec. T is the sample time. The justification for these particular time constant values will be provie shortly. The use of two 1st orer filters in series reuces estimate error cause by ifferentiating high-frequency coe measurement noise when compare to a single 1 st orer filter. This effect is clearly emonstrate in Figure 3 an Figure 4, which show filter outputs to simulate white noise with stanar eviation of 0.5 m along with the theoretically erive stanar eviation envelopes. It is clear from these two figures that the performance of the n orer implementation with a 30 sec time constant is superior (lower output noise) to a 1 st orer implementation, even when the latter has a much longer filter time constant (00 sec). The ability to use a shorter time constant will result in quicer etection of CCD failures. Following the ivergence estimator is the etection function, which is a simple threshol test: If >T cc then alarm, else no alarm. The etection threshol efine as: T = σ (3) cc ff, mon 486

5 1) Nominal Ionospheric Divergence Contribution: The nominal value for ionospheric ivergence use in the LGF specification an the MOPS ( σ = m/s) was selecte i to ensure that the protection levels compute at the aircraft woul be conservatively large. However, assuming such a large value for σ for the CCD monitor will cause i unreasonably loose threshols. In fact, prior research [10] suggests that a more realistic nominal value is σ i m/s. Archive ual frequency carrier phase ata from the LTP was use to substantiate this result, as is iscusse below. Figure 3. 1 st Orer Filter (ˆ response) τ 1 = 00sec. Ashtech receiver L1/L carrier phase ata was use to compute instantaneous ionospheric ivergence rates over 1 sec measurement intervals with a 5 eg elevation mas. The instantaneous rates were then average in 100 sec winows to reuce the ifferentiate carrier phase noise contribution to the rate estimates. The analysis was carrie out for 17 satellites over 7 months with ata taen from one ay in each of these months (see Table 1). Figure 5 shows an example ivergence trace for a single satellite on a single pass. Table 1. Nominal Ionospheric Divergence Data Archive SV # 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 0 Date (Year 004) Feb 11, Mar 11, Apr 16, Jun 15, Jul 15, Aug 31, an Oct 05 Figure 4. Two 1st Orer Filters in series ( τ1 = τ = 30sec ) In equation (3), σ is the fault-free stanar eviation of the test statistic an ff,mon is a constant chosen to ensure that the probability of fault-free alarm meets the allocate continuity requirement for the monitor [3]. Both σ an ff,mon are algorithm parameters, whose values are nominally set at σ = m/s an ff,mon = The justification for σ value is given in the next sub-sections. A. Fault-Free Distribution: As state above, in orer to set the monitor threshol, we nee the fault-free stanar eviation ( σ ) of test statistic. This fault-free istribution will be affecte by both filtere coe noise ( σ n ) an nominal ionospheric ivergence ( σ i ) which are assume to have istributions N(0, σ n ) an N(0, σ i ) respectively, such that: n(, τ 1, τ ) i σ = σ EL + σ where, EL is the satellite elevation. We will first consier the nominal ionospheric contribution. (4) Figure 5. Example Divergence trace: SV#3 Feb Figure 6 shows the cumulative istribution function (CDF) of all of the empirical ivergence rates from the entire ata set. The ashe line in the plot is the CDF of a gaussian istribution with the same stanar eviation as the ata ( σ i = m/s). It is clear that the ionospheric ivergence rate istribution is not gaussian in the istribution tails. However, Figure 7 shows that when the original stanar eviation is inflate by a factor of.85, the new gaussian istribution oes boun the empirical CDF. 487

6 σ i = m/s Figure 6. CDF Plot for Combine LTP Divergence Data Archive fiel test ata provie by Honeywell International (LGF contractor) was use to etermine σ n as a function of EL for ifferent values of τ. Data from the Honeywell LGF receivers equippe with Multipath Limiting Antennas (MLA) an High Zenith Antennas (HZA) was use as the source for coe an carrier ata as input into the ivergence rate estimator in equation (). To isolate the coe noise contribution, carrier phase ata from a nearby ual frequency Novatel OEM4 receiver was use to remove nominal ionospheric ivergence from the ata prior to processing. Table provies a summary of the empirical results for σ n in m/sec. It is clear from the results that the resulting σ n is an orer of magnitue smaller than σ i. The contribution of σ n to σ is negligible even for τ = 0 sec relative to the effect of σ i. Figure 8 shows the stanar eviation of filtere coe noise against time constant for a single satellite. SV 13 Jul HZA Data σ i = m/s Figure 7. Gaussian Overboun of LTP Ionospheric Divergence Rate Data. Thus, the value to be use is σ i = = m/s.. Filtere Coe Noise Contribution: As inicate in equation (4), the contribution of filtere coe noise (σ n ) epens on satellite elevation (EL) an the filter time constants τ 1 an τ. In principle, τ 1 an τ can always be chosen large enough to mae σ σ i (although care must be taen to ensure that the time to etect oes not become too large). For simplicity in the monitor implementation an analysis, we efine τ =τ 1 =τ. Figure 8. HZA Divergence Estimate error vs. Time constant For this initial monitor analysis a nominal value for τ = τ 1 = τ = 30 sec is selecte. Table. σ n (m/s) vs. Elevation an Time Constant EL τ 5-15 (MLA) (MLA) (HZA) (HZA) (HZA) >60 (HZA) 0sec sec sec

7 Monitor Definition Value Parameter T Sample Time 0.5sec ff,mon Constant 5.83 τ Time Constant 30sec σ st. ev. of m/s (fault free test metric) Having set the nominal CCD monitor parameters, which are summarize in Table 3, we next procee to the integrity analysis. IV. General Integrity Analysis for Space Segment Failure The following efinitions will be use below in the erivation of the probability of Loss of Integrity (LOI) ue to a general space segment failure event on satellite : e v Vertical component of ifferential position error ue to all sources v i Differential ranging error for satellite i ue to all fault-free error sources σ i Stanar eviation of v i b Differential ranging error on satellite ue to satellite failure only n S S v,i Table 3. Monitor Moifiable Parameters Number of satellites in view Weighte pseuoinverse use at aircraft for position fix (which is a function of satellite geometry) The element of matrix S projecting the ranging measurement from satellite i into the vertical irection q Test statistic for the fault monitor for satellite ue to all sources r Test statistic for the fault monitor for satellite ue to the satellite failure only η Test statistic for the fault monitor for satellite ue to all fault-free error sources σ η, Stanar eviation of the monitor test statistic in the fault-free case σ r, Stanar eviation of the monitor test statistic for faulte case ff,mon The multiplier on σ η, use to efine monitor threshol with esire fault free etection probability Q Cumulative istribution function for the stanar normal istribution n i= 1 S v v, i i Vertical component of ifferential position error for all fault-free error sources n σ v Stanar eviation of S v, iv i = 1 ffm 5.81, the multiplier on σ r use to compute LAAS VPL H0 In this analysis, the probability of Loss of Integrity efine as: P(LOI fault ) P( e v > VPL fault ) P( q < ff,mon σ r, fault ) i (5) Consier the two terms on the right-han sie of equation (5) separately. Given a failure on satellite, with a resulting b not close to zero, the first term on the right han sie can be simplifie as a one-sie probability (Figure 9) given by: n P( ev > VPL fault ) P( Sv,b + Sv,ivi > ffmσv ) i= 1 = 1 Q{ ffmσv Sv, b } Sv,b = 1 Q ffm σv σv, b 1 Q = ffm. σv σ The worst-case probability occurs when σ v, 1. Therefore, σ v it is convenient to conservatively use the following satellite- Figure 9. LGF Integrity Ris given fault on Ranging Source geometry-free expression, b P( e VPL fault ) 1 Q v > = ffm σ b Q = ffm (6) σ Using the same metho the secon term on the right-han sie of equation (5) can be simplifie as follows: P( q < ff,mon σ r fault ffm σ v S v, b ) P( r + η < r = Q ffm,mon σ r, 1 cf ff,mon σ r fault ) (7) 489

8 Substituting the results (6) an (7) in to equation (5) yiels, P( LOI fault ) b Q σ = ffm r Q ff, mon σr, In general, a space segment failure event on satellite will cause ifferent transient responses in the ifferential position error b an the monitor test statistic r. The loss of integrity probability in equation (8) will be a function of both of these failure response functions, the failure magnitue, the elapse time since failure onset, an the groun an airborne receiver tracing start times (which influence σ an σ ). For r, every type of satellite failure it is necessary to fin the conitions that maximize the LOI probability. It is also necessary to etermine whether the LOI probability excees the integrity ris allocation for a failure moe for uration greater than the maximum permitte time-to-alert. For LGF integrity monitors, the require time-to-alert is 3 sec. A. Application to CCD Monitoring: In the section the goal is to obtain the terms in equation (8), when a failure occurring at the ranging source causes a CCD fault. In orer to o so we efine the following: t 0g LGF filter start time t 0a Airborne filter start time nom = m/s, the nominal ionospheric ivergence rate e g, Ranging error ue to nom at LGF relative to LGF steay state e a, Ranging error ue to nom at aircraft relative to LGF steay state (8) nominal ionospheric ivergence rate, then e g = σ iv_gn an e a = σ iv_air. Therefore, σ = σ σ (11) an combining with (9), b σ = nom pr_ gn + pr_ gn + eg ( t t0g) + ea ( t t0 a) e ( t max[ t σ g pr _ gn + σ 0 g pr _ air,0]) e ( t max[ t + e ( t t g a 0 g 0a,0]) ) + e ( t t a ) 0a (1) Equation (1) may be substitute into the first term on righthan sie of equation (8) for the CCD case. Ieally, the LGF monitor performance shoul be inepenent of σ pr_gn,nom an σ pr_air,nom, which may vary over time an location. In the most conservative analysis, it is assume that σ pr_gn,nom = σ pr_air,nom = 0. As iscusse earlier, the LGF filter is a first orer igital LTI filter with a 100 sec time constant (efine in the LGF Specification). In the current LGF prototype implementation, there is no correction broacast for satellites uring the first 00 sec of filtering (i.e., only t t 0g > 00 sec nees to be consiere in the integrity analysis). For the aircraft, as iscusse in section II, it is assume a first orer igital LTV filter is implemente. This filter iffers from the LGF filter only uring the first 100 sec of operation, when the effective filter time constant increases uniformly in time (up to the 100 sec limit). It is also assume that the aircraft will use filtere measurements immeiately (i.e., t t 0a > 0 nees to be consiere in the integrity analysis). The two igital filter responses to nominal ivergence, e g an e a, are plotte as function of time in Figure 10. Given a ivergence failure with a CCD rate an time of onset t = 0, a ifferential ranging error exists only when both filters are tracing (t > t 0a an t > t 0g ) an after onset of the failure (t > 0). Therefore the ifferential ranging error, b in the general analysis in the preceing section, can be expresse as b = nom eg (t max[t0g,0]) ea(t max[t0a,0]) (9) Note that t 0g an t 0a can be negative, signifying possible filter start times prior to failure onset. The stanar eviation of the ranging error use at the aircraft, σ in the preceing section, is σ = σ pr_gn + σiv_gn + σ pr_air + σiv_air (10) As nom = m/s is prescribe by the LAAS MOPS an LGF Specification to be use as the stanar eviation of Figure 10. Differential ranging error relative to LGF steay state For the CCD monitor, the ivergence rate estimate is the test statistic r in the general analysis in the preceing section. Therefore, 490

9 (t) r = m/s σ r, 0 t > t 0g t < t 0g (13) This may be substitute into the secon term on the right-han sie of equation (8) for the CCD case. The example noise-free time response of to a unit ramp ivergence input, = 1, is shown in Figure 11 for the igitally implemente estimator. Because the ivergence estimator is a linear filter, the amplitue of the time response for other ivergence inputs will simply scale linearly with. If the groun filter starts close in time to the aircraft filter, then the monitor response for the first 00 secons is not utilize for computation of P m fault because the LGF oes not broacast corrections uring this time. Hence P m fault = 0 uring the first 00 sec. Therefore, as illustrate in Figure 1 the worst-case P(LOI fault ) occurs when the groun filter has starte well before the aircraft filter: theoretically speaing, when t 0g = - an aircraft filter has just starte. This result is further illustrate in Figure 13, which shows contours of the highest values of P(LOI fault ) for any value of t at specifie values of t 0g an t 0a. The highest values of P(LOI fault ) occur where in t 0g is at its lowest value ( 500 sec in the figure). =0.03m/s Figure 11. Noise free Divergence Estimate for =1m/s. B. Integrity Analysis Results: The ivergence onset of magnitue is efine to occur at time t = 0. For the purpose of interpreting the integrity analysis results, we will refer to the first term on the right-han sie of equation (8) as P ev fault (t, t 0a, t 0g, ) an the secon term as P m fault (t - max [t 0g, 0], ). Figure 13. Worst-case P LOI fault with fixe. Next, we use the above result to calculate P(LOI fault ) with t 0g fixe at a value far behin t 0a, an vary t, t 0a an. Figure 14 shows a 3-D plot of the resulting values of P(LOI fault ). An interesting point to note from this plot is that as the fault magnitue () increases P(LOI fault ) is worst when t 0a is small an time (t) approaches t 0a,. 00sec wait P (ev>vpl) fault Air LGF t 0g 0 t 0a t (m/s) T cc P m fault t 0 Figure 1. Illustration for P (ev fault) an P m fault Figure 14. Worst case P LOI fault To explain this result, consier again Figure 1 were the LGF filter starts well before the ivergence onset (at t = 0) an the aircraft filter starts after the fault onset. Now, P m fault improves as time increases. Hence, P m fault is worst when the monitor response to fault has just starte. i.e. for small t. On the other 491

10 han, P ev fault gets worse as t t 0a an then improves for large values of t as the two filters approach the same steay state value. Therefore, the worst-case P(LOI fault ) as shown in Figure 14 occurs at small values of t 0a with t=t 0a. The total specifie Time to Alert (TTA) for LAAS is 6 secons, with 3 secons allocate to the groun an 3 secons for air. The preliminary TTA analysis in this paper oes not yet aress the issues relate to transmission elays of signal if a fault is etecte. In orer to carry out a preliminary TTA analysis for ifferent fault magnitues, we chec the instances in which P(LOI fault ) >10-4 (the specifie maximum probability of MI given the CCD fault) for each value of, an the uration of such occurrences to verify if it is uner 3 sec (allocation for groun). Figures 15 an 16 show the results. Figure 15 shows the worst-case value (for any t, t 0g, an t 0a ) of P (LOI fault) an the uration of time this probability excees 10-4 It is clear from this figure, that there exist some occurrences of time in LOI of 3.5 sec, which excees the groun allocation to the time-to-alert (3 sec). The figure clearly shows that the result is on the ege of meeting the time-to alert requirement. This observation is further supporte by the fact that the situation is easily remeie by reucing the monitor filter time constants from 30 sec to 9 sec. The result is shown in Figure 16. Figure 16. CCD LTI/LTV P(LOI) an Time in LOI V. Conclusion In this paper we have aresse the monitoring an integrity ris analysis for coe minus carrier ivergence GPS satellite faults. The paper suggests a 1 st orer LTV smoothing filter as a goo choice for implementation at the aircraft an focuses the analysis on this implementation, but as other higher filter implementations are also possible the analysis approach was esigne to be easily extenable to ifferent filter implementations at the aircraft. A CCD monitor to be implemente at the LAAS groun facility was esigne. This monitor uses two 1 st orer LTI filters in series for ivergence rate estimation, which is followe by a simple etection test. The paper gives the nominal values for monitor filter time constants. The etection threshol is set by computing the fault free test metric base on experimental ata. We also present a new irect approach to compute the probability of Loss of Integrity for a space segment failure an apply it to the CCD monitoring problem. Different aircraft an groun filter start times are explicitly accounte for. Preliminary analysis results show that LAAS integrity requirements are satisfie. Figure 15. CCD LTI/LTV P(LOI) an Time in LOI However, even this minor reuction in the filter time constant may not be necessary. The analysis is conservative because fault-free ionospheric ivergence ranging errors are implicitly inclue in v i in the evelopment if equation (6). In reality, these errors will contribute irectly to the ivergence failure i.e., slightly changing the effective value of. The result is that the probability in equation (6) is conservatively compute in this analysis. A substantiation of these claims an a more etaile analysis of time-to-alarm problem will be the subjects of a future paper. Acnowlegements The authors gratefully acnowlege the Feeral Aviation Aministration Satellite navigation LAAS Program Office for supporting this research. We also than Honeywell International for proviing LGF fiel ata that was use in this wor. However, the views expresse in this paper belong to the authors alone an o not necessarily represent the position of any other organization or person. 49

11 References [1] Shively, C., Derivation of Acceptable Error Limits for Satellite Signal Faults in LAAS, Proceeings of ION GPS-99, September [] Zaugg, T., A New Evaluation of Maximum Allowable Errors an Misse Detection Probabilities for LAAS Ranging Source Monitors, Proceeings of ION 58 th Annual Meeting, June, 00, Albuquerque, NM. [3] Cassell, R., Derivation of LAAS Category I PSP Monitor Threshols, Report to FAA William J. Hughes Technical Center, September 16, 005. [4] FAA-E-937A, Performance Type One Local Area Augmentation System (LAAS) Groun Facility Specification, April 17, 00. [5] DO-53A Minimum Operational Performance Stanars for the Local Area Augmentation System Airborne Equipment, RTCA, November 8, 001. [6] Rife, J., Formulation of a Time-Varying Maximum Allowable Error for Groun-Base Augmentation Systems, Proceeings of the ION National Technical Meeting, January 006, Monterey, CA. [7] Shively, C., Ranging Source Fault Integrity Concepts for a Local Airport Monitor for WAAS, Proceeings of the ION National Technical Meeting, January 006, Monterey, CA. [8] Rife, J., Pervan, B. Unpublishe presentation Time- Varying MERR Applie to CCD, September 005. [9] Pervan, B., an Simili, D., Algorithm Description Document for the Coe-Carrier Divergence Monitor of the Local Area Augmentation System, August 5, 005. [10] Christie.J, et al., The Effects of Local Ionospheric Decorrelation on LAAS: Theory an Experimental Results, ION National Technical Meeting, January